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International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
May 25, 2015 14:55, Ôóíêöèîíàëüíûå ïðîñòðàíñòâà, Moscow, Steklov Mathematical Institute of RAS

My Japanese book «Theory of Besov spaces, including a remark on the space $S'$ over $P$»

Y. Sawano

Tokyo Metropolitan University

Abstract: Let ${\mathcal S}'$ denote the set of all Schwartz distributions and ${\mathcal P}$ the set of all polynomials. If we define ${\mathcal S}_\infty$ to be the set of all $f \in {\mathcal S}$ such that $\int_{{\mathbb R}^n}x^\alpha f(x) dx=0$ for all $\alpha$, we can consider the dual space ${\mathcal S}_\infty'$.
We know that ${\mathcal S}_\infty'$ is isomorphic to ${\mathcal S}'/{\mathcal P}$ as linear spaces. But it seems to me that this is true topologically. In my Japanese book, I wrote a proof but I have commited the mistake. But recently I modified the proof. My result is as follows.
Theorem. Equip ${\mathcal S}'$ and ${\mathcal S}'_\infty$ with the weak star topology. Then the restriction mapping from ${\mathcal S}'$ to ${\mathcal S}_\infty'$ is open.

Materials: abstract.pdf (84.9 Kb)

Language: English

References
1. S. Nakamura, T. Noi, Y. Sawano, “Generalized Morrey spaces and trace operator” (to appear)
2. Y. Sawano, Theory of Besov Spaces, Nihon-Hyoronsha, 2011, 440 pp. (in Japanese)

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